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Title:Linear and Nonlinear Stability Analyses of Density-Wave Oscillations in Heated Channels
Author(s):Uddin, Rizwan
Doctoral Committee Chair(s):Dorning, J.J.,
Department / Program:Nuclear Engineering
Discipline:Nuclear Engineering
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Engineering, Nuclear
Abstract:Linear and nonlinear stability analyses of density-wave oscillations in uniformly and non-uniformly heated channels are reported. The two-phase flow is represented by the drift flux model. A constant characteristic velocity v$\sbsp{\rm o}{\*}$ is used to make the set of equations dimensionless to ensure the mutual independence of the dimensionless variables and parameters.
The exact equation for the total channel pressure drop for the uniform heat flux case is perturbed about the steady-state for the linear and nonlinear analyses. The surface defining the marginal stability boundary (MSB) is determined in the three-dimensional equilibrium-solution/operating-parameter space v-N$\sb{\rm sub}$-N$\sb{\rm pch}$. The effects of the void distribution parameter C$\sb{\rm o}$ and the drift velocity V$\sb{\rm gj}$ on the MSB are examined.
The nonlinear analysis carried out using the Lindstedt-Poincare technique, shows that supercritical Hopf bifurcation occurs for the regions of parameter space studied; hence stable oscillatory solutions exist in the linearly unstable region in the vicinity of the MSB. That is, the stable fixed point bifurcates to unstable fixed point and stable limit cycle as the MSB is crossed by varying N$\sb{\rm sub}$ and/or N$\sb{\rm pch}$.
The Lindstedt-Poincare technique for the more general non-uniform heat flux case is applied directly to the set of nonlinear partial differential equations. The characteristic equation obtained via linear analysis is solved for the MSB's. The effects of the heat flux shape on the marginal stability boundary are reported.
The nonlinear analysis for the non-uniform heat flux case also leads to supercritical Hopf bifurcation and the oscillation amplitude, as a function of the distance from the MSB, rises faster for channels with heat flux shapes that lead to an overall less stable channel (linear analysis) compared to an equivalent uniformly heated channel.
The nonlinear dynamics of two-phase flow in uniformly heated chemicals has also been studied numerically. The set of two nonlinear functional ODEs obtained for the dynamics of the heated channel with two-phase flow was integrated numerically for parameter values in different regions of the parameter space and various initial conditions. Stable limit cycles exist for parameter values in the linearly unstable region close to the density-wave marginal stability boundary (MSB), and the oscillation amplitude grows monotonically with the distance from the MSB. Sensitive dependence upon initial histories (t $<$ 0) and initial conditions (t = 0) is found for parameter values for which there exist more than one attracting set--stable fixed points, stable limit cycle.
Issue Date:1987
Description:255 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987.
Other Identifier(s):(UMI)AAI8803224
Date Available in IDEALS:2014-12-16
Date Deposited:1987

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